We argue that four-dimensional quantum gravity may be essentially renormalizable provided one relaxes the assumption of metricity of the
theory. We work with Plebanski formulation of general relativity in which the metric (tetrad), the connection as well as the curvature are all independent variables and the usual relations among these quantities are only on-shell. One of the Euler-Lagrange equations of this theory guarantees its metricity. We show that quantum corrections generate a counterterm that destroys this metricity property, and that there are no other counterterms, at least at the one-loop level. There is a new coupling constant that controls the non-metric character of the theory. Its beta-function can be computed and is negative, which shows that the non-metricity becomes important in the infra red. The
new IR-relevant term in the action is akin to a curvature dependent
cosmological ``constant\'\' and may provide a mechanism for naturally small ``dark energy\'\'.